Question

cosec 48° + cosec 96° +
cosec 192° + cosec 384° = 0)​

Answer 1

Answer:

> cosec 48 = cosec (90 - 42)

=> cosec48 = sec 42

=> cosec48 = 1/cos 42

_______

and

=> cosec 192 = cosec (270 - 78)

=> cosec192 = -sec 78

=> cosec192 = -1/cos 78

_______

So,

=> cosec 48 + cosec 192

= 1/cos 42 - 1/cos 78

=> cosec48 + cosec192

=(cos 78 - cos 42)/(cos 78 × cos 42)

= -(2×sin 60 × sin 18)/{cos (60 + 18) × cos (60 - 18)} {Apply cos C - cos D formula}

= -(2× sin 60 × sin 18)/{cos2 60 - sin2 18}

= -(2× √3/2 × (√5 - 1)/4}/{(1/2)2 - ((√5 - 1)/4)2 }

= -2√3

__________

Again,

Now,

=> cosec 384 = cosec (360 + 24)

=> cosec384 = cosec 24

_________

So,

=> cosec 96 + cosec 24

= 1/sin 96 + 1/sin 24

= (sin 96 + sin 24)/(sin 96 × sin 24)

= (2× sin 60 ×cos 36)/{sin (60 + 36) × sin (60 - 36)} {Apply cos C - cos D formula}

= (2×sin 60 ×cos 36)/{sin2 60 - sin2 36}

= (2× √3/2 ×(√5 + 1)/4}/{(√3/2)2 - ((√5 + 1)/4)2 }

= 2√3

__________

So,

=> cosec 48 + cosec 96 + cosec 192 + cosec 384

=> -2√3 + 2√3 = 0

_______________[PROVED